The Big Idea: What Is Proportion?

Proportion describes a predictable relationship between two quantities. When one changes, the other changes in a consistent way.

There are two types:

Type	What Happens	Key Test	Equation
Direct	Both increase (or decrease) together	Ratio $y/x$ is constant	$y = kx$
Inverse	One increases, the other decreases	Product $xy$ is constant	$y = k/x$

The number $k$ is called the constant of proportionality — it’s the glue that holds the relationship together.

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Part 1: Direct Proportion

What Does “Directly Proportional” Mean?

When $y$ is directly proportional to $x$ (written $y \propto x$), it means:

• Double $x$ → $y$ doubles
• Triple $x$ → $y$ triples
• Halve $x$ → $y$ halves

The ratio $\frac{y}{x}$ stays the same no matter what values you pick.

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How to Recognize Direct Proportion

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The 3-Step Method: Finding the Equation

Every direct proportion exam question follows this pattern:

Step 1: Write $y = kx$

Step 2: Substitute known values to find $k$

Step 3: Write the final equation and solve

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Graphs of Direct Proportion

The graph of $y = kx$ is always a straight line through the origin (0, 0).

Feature	What to Look For
Shape	Straight line
Passes through	Origin (0, 0) — always
Gradient	Equal to $k$ (the constant of proportionality)
Steeper line	Larger $k$ value

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Direct Proportion with Squares: $y \propto x^2$

Sometimes $y$ is proportional to the square of $x$, not $x$ itself.

$y \propto x^2 \implies y = kx^2$

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Part 2: Inverse Proportion

What Does “Inversely Proportional” Mean?

When $y$ is inversely proportional to $x$ (written $y \propto \frac{1}{x}$), it means:

• Double $x$ → $y$ halves
• Triple $x$ → $y$ becomes one-third
• Halve $x$ → $y$ doubles

The product $x \times y$ stays the same no matter what values you pick.

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How to Recognize Inverse Proportion

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The 3-Step Method for Inverse Proportion

Same structure, different equation:

Step 1: Write $y = \frac{k}{x}$ (or equivalently, $xy = k$)

Step 2: Substitute known values to find $k$

Step 3: Write the final equation and solve

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Graphs of Inverse Proportion

The graph of $y = \frac{k}{x}$ is a curve called a hyperbola.

Feature	What to Look For
Shape	Smooth curve (NOT a straight line)
Passes through origin?	No — never
Approaches axes	Gets closer and closer but never touches (asymptotes)
As $x$ increases	$y$ decreases toward zero

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Inverse Square: $y \propto \frac{1}{x^2}$

Just like direct proportion has $y \propto x^2$, inverse proportion has $y \propto \frac{1}{x^2}$.

$y \propto \frac{1}{x^2} \implies y = \frac{k}{x^2}$

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Part 3: The Unitary Method (Quick Shortcut)

For simple problems, you can skip the equation and use the unitary method: find the value for 1 unit, then scale.

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Part 4: Direct vs Inverse — How to Tell

This is a common exam question: given a scenario, identify which type of proportion applies.

Scenario	Type	Why
More workers → less time	Inverse	Product (workers × time) is constant
More hours → more distance	Direct	Ratio (distance/hours) is constant
More taps → less time to fill	Inverse	Product (taps × time) is constant
More fabric → more cost	Direct	Ratio (cost/fabric) is constant
Faster speed → less time	Inverse	Product (speed × time) = distance
More people sharing → less each	Inverse	Product (people × share) is constant

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5 Common Mistakes That Cost Marks

Mistake 1: Using the Wrong Proportion Type

Wrong: “More workers finish faster, so time is directly proportional to workers.”

Right: More workers → less time. This is inverse proportion.

Fix: Always ask: “Do they move in the same direction or opposite?”

Mistake 2: Forgetting to Square (or Square Root)

When the question says ”$y$ is proportional to the square of $x$,” students write $y = kx$ instead of $y = kx^2$.

Fix: Underline the exact proportionality phrase in the question before writing the equation.

Mistake 3: Mixing Up the Tests

• Direct: check the ratio ($y/x$ = constant)
• Inverse: check the product ($x \times y$ = constant)

Using the wrong test will give you the wrong conclusion.

Mistake 4: Writing $y = kx$ for Inverse Proportion

The equation for inverse proportion is $y = \frac{k}{x}$, NOT $y = kx$. This mistake leads to completely wrong answers.

Mistake 5: Not Verifying Your Answer Makes Sense

After solving, do a quick sanity check:

• Direct: if $x$ increased, did $y$ increase too?
• Inverse: if $x$ increased, did $y$ decrease?

If your answer contradicts this, you’ve made an error somewhere.

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Quick Reference Card

Concept	Direct Proportion	Inverse Proportion
Symbol	$y \propto x$	$y \propto \frac{1}{x}$
Equation	$y = kx$	$y = \frac{k}{x}$
Test	$\frac{y}{x} = k$ (constant ratio)	$xy = k$ (constant product)
Graph	Straight line through (0, 0)	Curved hyperbola
Effect of doubling $x$	$y$ doubles	$y$ halves
Squared version	$y = kx^2$	$y = \frac{k}{x^2}$

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Try It Yourself

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